Near-Field Generation and Control of Ultrafast, Multipartite Entanglement for Quantum Nanoplasmonic Networks

For a quantum Internet, one needs reliable sources of entangled particles that are compatible with measurement techniques enabling time-dependent, quantum error correction. Ideally, they will be operable at room temperature with a manageable decoherence versus generation time. To accomplish this, we theoretically establish a scalable, plasmonically based archetype that uses quantum dots (QD) as quantum emitters, known for relatively low decoherence rates near room temperature, that are excited using subdiffracted light from a near-field transducer (NFT). NFTs are a developing technology that allow rasterization across arrays of qubits and remarkably generate enough power to strongly drive energy transitions on the nanoscale. This eases the fabrication of QD media, while efficiently controlling picosecond-scale dynamic entanglement of a multiqubit system that approaches maximum fidelity, along with fluctuation between tripartite and bipartite entanglement. Our strategy radically increases the scalability and accessibility of quantum information devices while permitting fault-tolerant quantum computing using time-repetition algorithms.

which may be compensated for by adjusting the input power, and therefore for consistency we use the same input field throughout the study. We then separately solve for the time-dependent density matrix of the 3-qubit system for spin-½ QDs (88 matrix) and remove the waveguide core and MIM resonator from numerical simulations considering the incident electric field as their primary contribution to the dynamics of the QD system. This method has been effectively used to model the power dissipation and heat transfer dynamics in similar NFT + media systems.
where Γ is the super operator that includes the decay and dephasing of each energy level. Here, we consider relatively fast decay/dephasing rates with dephasing four times that of the decay.
Although they are not required, lower temperatures are known to improve these values. The Hamiltonian shown below uses raising and lowering energy level operators, σ ± , for a 3-qubit ( ∈ 1 − 3) system, with ω the frequency of the QD energy level. Of note, a coherent coupling term between quantum emitters is not explicit in the Hamiltonian, rather it is implicitly incorporated by the spatial integration of the displacement field throughout the media.
We use a symmetry-adapted averaging approach of the electric field to exclude all self-interactions the QD may have with its emitted field S2 . Therefore, the scattered field used in the Hamiltonian is defined as an average of the full field solution in the region of the QD, E sc = 1 ∑ (r ), where 'N' is the number of data points defined within the QD with the summation going from 1 to N.
The FETD simulations utilize a free tetrahedral mesh (Shown in the supplementary information) with distances between data points ranging from 0.6 nm -32 nm, maximum element growth rate of 1.5, and relative tolerances tested between 0.001-0.01. Density matrix elements are calculated with an error estimated on the order of 10 -3 . A modestly shared coherence between states is found suitable for initialization of the system (See section on initial conditions).
The density matrix is formed by taking the tensor product for three, 2-level quantum dots (QD) defined by a wave function using the standard basis state, |Ψ〉 = α|0〉+β|1〉, for individual QDs with αα * and ββ * the probabilities to be in either the ground or excited state, respectively. The tensor product is defined as |φ〉 = |Ψ 1 〉 ⊗|Ψ 2 〉⊗ |Ψ 3 〉 and the density operator as ̂= | ⟩⟨ | for the tripartite system. This yields an 8×8 matrix when applying the appropriate bra and ket notation with elements corresponding to the tripartite basis states |000⟩, |010⟩, |001⟩, |100⟩, |101⟩, |110⟩, |011⟩, |111⟩. The diagonal elements yield the populations of each level while the off-diagonal elements are the coherences that are necessary to calculate the polarization and fidelity. We then solve the density matrix equations while ensuring the trace ∑ ρ 4 =1 = 1 is conserved. The incident electric field profile is considered monochromatic and time dependent which yield the detunings, , between each energy level and the laser. Parameters optimized for the tripartite system are reported in Tables S1 and S2 while Figure S7 shows a to-scale schematic of the full photonic waveguide and NFT structure.
In addition to using the fidelity of the GHZ state to calculate tripartite entanglement, there is a condition for the coherence terms to generate genuine multipartite entanglement (GME) from a biseparable state as defined in Gühne et al S3 and given by | 18 | > √ 22 77 + √ 33 66 + √ 44 55 .
Satisfying this condition is equivalent to achieving tripartite entanglement, which we have plotted with the fidelity in Figure 3c of the main text, though we adapt the condition to the form, in order to match when the fidelity goes over 0.5. Values over 0.5 for fidelity demonstrate multiparticle entanglement, which the GME condition corroborates, with bipartite entanglement achieved between two states of the 3-qubit density matrix for values under 0.5 S4 .

Initial Conditions
The initial states used in our simulations are reported in the charts of Figure S1 that yield an initial fidelity of roughly 0.4, i.e. no initial multipartite entanglement. We find nothing particular about the initial conditions we use to produce tripartite entanglement except to have a shared coherence between coherence terms. In our case, the off diagonal elements have a modest coherence with amplitudes on the order of 0.2 or less that may be experimentally realized S5 . Tripartite entanglement has also been successfully reproduced by setting the imaginary terms to zero, and even by further reducing the real terms by at least 10% for certain conditions. We have found these requirements on the shared coherence by setting an initial condition with tripartite entanglement and watching the system evolve in time to an unentangled 3-qubit state.
Although no magnetic transitions were utilized in our study, we highlight the magnetic fields that would otherwise be incident on the QD media in Figure S2. We note that components of the magnetic field may be enhanced and magnetic transitions, such as those in Diamond vacancy centers, can be used as an alternative quantum emitter.  convergence of Maxwell's equations and the density matrix. As we move away from the QD positions asymmetries in the mesh arise, which is typical when using conformal meshes which depends on mesh (tetrahedral in our case) and overall structure shape. Though it should be noted the field profile and shape matches well to previous simulations using symmetric meshes.
To solve for the maximum temperature of the Au within the NFT, we calculate the steady-state form of the thermal diffusion equation, defined as The contours can be seen to outline the boundary of the SiO2 insulator layer. (b) The temperature is shown to be a maximum in the media with an estimated temperature of 400 K predicted in the Au. One can manipulate the temperature by adding heatsinking material or using metallic alloys in the NFT.

Temperature Calculations
where is the diffusion coefficient and the heat source, , is considered to be dominated by the resistive heating, i.e. power loss, in the system which we extract from the steady-state solution of Eq. S5 and shown in Figure S8.    Table S3.